[gryn]

because people don't know what effects that would have on a growing baby. (I'm not saying scientists but just random people)

questions that would worry them would probably be things like what if the baby start depending on them and their bodies reduce/ceases to produce melatonin naturally, and that this permanently affects the baby.

I haven't looked into the findings for theses kind of questions, but even if you look into it you'll also have to be condident of the reliability of the results. maybe it's harmless for adults but not for 2 year olds. Maybe the pediatrician you went to see didn't prescribe them for a reason.

Edit: a quick Google search in English mostly found positive articles about it, while the same search in french found negative articles warning about the potential risks like this one: https://www.vidal.fr/actualites/29656-melatonine-non-sans-da...